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Theorem grpideu 17794
Description: The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)
Hypotheses
Ref Expression
grpcl.b 𝐵 = (Base‘𝐺)
grpcl.p + = (+g𝐺)
grpinvex.p 0 = (0g𝐺)
Assertion
Ref Expression
grpideu (𝐺 ∈ Grp → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
Distinct variable groups:   𝑥,𝑢,𝐵   𝑢,𝐺,𝑥   𝑢, + ,𝑥   𝑥, 0
Allowed substitution hint:   0 (𝑢)

Proof of Theorem grpideu
StepHypRef Expression
1 grpmnd 17790 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
2 grpcl.b . . 3 𝐵 = (Base‘𝐺)
3 grpcl.p . . 3 + = (+g𝐺)
42, 3mndideu 17664 . 2 (𝐺 ∈ Mnd → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
51, 4syl 17 1 (𝐺 ∈ Grp → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  ∃!wreu 3119  cfv 6127  (class class class)co 6910  Basecbs 16229  +gcplusg 16312  0gc0g 16460  Mndcmnd 17654  Grpcgrp 17783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015  ax-pow 5067
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786
This theorem is referenced by: (None)
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